US distress models revisited

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Introduction

Hazard models are powerful distress models.

Popular since Shumway (2001).

\[\begin{align*} y_{it} &\in\{0,1\}\\ g(E(y_{it}|\vec x_{it}, \vec z_t)) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t \end{align*}\]

\(y_{it}\)s are binary distress indicator, \(\vec x_{it}\)s are firm covariates, and \(\vec{z}_t\)s are macro variables.

E.g., see Chava and Jarrow (2004), Beaver, McNichols, and Rhie (2005), and Campbell and Szilagyi (2008).

Large time variation

Line is a generalized additive model with a spline for the intercept (Wood 2017).

Cannot be captured by firm specific covariates, \(\vec x_{it}\), and macro covariates, \(\vec z_t\). Major issue for portfolio risk.

Frailty models

Extend to include latent effects

\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, \vec u_t, \vec o_{it})) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + \vec u_t^\top \vec o_{it} \\ \vec u_t &\sim p_{\vec \theta}(\vec u_{t-1}) \end{align*} \]

Pioneered by Duffie et al. (2009).

E.g., see Koopman, Lucas, and Schwaab (2011), Hwang (2012), Chen and Wu (2014), Qi, Zhang, and Zhao (2014), Nickerson and Griffin (2017), Azizpour, Giesecke, and Schwenkler (2018), and Kwon and Lee (2018) for other papers on time-varying effects.

Only a frailty model as in Vaupel, Manton, and Stallard (1979) for certain link functions where \(\vec u_t^\top \vec o_{it}\) is a multiplicative factor on the instantaneous hazard. This is not the case for e.g., the logit link function where \(\vec u_t^\top \vec o_{it}\) is a multiplicative factor on the odds.

Motivation

Although an OU process is a reasonable starting model for the frailty process, one could allow much richer frailty models. …, however, we have found that even our relatively large data set is too limited to identify much of the time-series properties of frailty. … For the same reason, we have not attempted to identify sector-specific frailty effects.

Duffie et al. (2009)

Talk overview

Data set.

Models without frailty.

Models with frailty.

Data set

Merge CRSP and Compustat.

Merge with Moody’s Analytics Default & Recovery Database.

Distress events

At risk: has not had a distress, is after resolution date, or 12 months after distress if resolution date is missing.

Events are alternative dispute resolution, conservatorship, cross default, loan forgiven, chapter 7, placed under administration, seized by regulators, deposit freeze, suspension of payments, grace period default, payment moratorium, prepackaged chapter 11, indenture modified, bankruptcy, receivership, missed principal payment, missed principal and interest payments, distressed exchange, chapter 11, and missed interest payment.

Missed interest payment is by far the most common.

Matching to Compustat is done with CUSIPs, ticker symbols, and company names.

Compustat data

Use quatlery data and impute yearly data when missing.

Data is lagged with 3 months and carried forward for up to 1 year.

CRSP

Used to compute idiosyncratic volatility. Standard deviation from 1 year rolling regression of daily stock log return on the log market return. Require 3 months data in window and 1 year initial data.

Used to compute past 1 year excess log return.

Used to compute log relative market size.

Data is lagged with 1 month and carried forward for up to 3 months.

Distance-to-default

Very good predictor (Bharath and Shumway 2008).

Computed with the so-called KMV method (Vassalou and Xing 2004). Computed with daily stock data, debt data from Compustat, the 1 year T-bill rate, and with a rolling 1 year window.

Require 1 year initial data and at least 3 months of data in each window.

Final sample

Exclude firms with first digit SIC code 6 or 9. Use Compustat siccd if available. Otherwise use CRSP sich.

Final samples has 624,692 firm-months, 4,433 firms and 721 events.

Models without frailty

Included variables are

  • Working capital / total assets
  • Retained Earnings / total assets
  • Operating income / total assets
  • Sales / total assets
  • Net income / total assets
  • Total liabilities / total assets
  • Current ratio
  • Market value / total liabilities
  • Idiosyncratic volatility
  • Log excess return
  • Relative log market size

All winsorized at 1% and 99%.

First model

Df AIC
First model 12 7612.93
New denominator 12 7559.93
Add distance-to-default 13 7469.90
Add macro variables 15 7447.24
Simplify model 11 7441.04

Change denominators

Df AIC
First model 12 7612.93
New denominator 12 7559.93
Add distance-to-default 13 7469.90
Add macro variables 15 7447.24
Simplify model 11 7441.04

Issue with small denominators as in Campbell and Szilagyi (2008). Change denominator to 50% total assets and 50% market value.

Add distance-to-default

Df AIC
First model 12 7612.93
New denominator 12 7559.93
Add distance-to-default 13 7469.90
Add macro variables 15 7447.24
Simplify model 11 7441.04

Macro variables

Df AIC
First model 12 7612.93
New denominator 12 7559.93
Add distance-to-default 13 7469.90
Add macro variables 15 7447.24
Simplify model 11 7441.04

Past 1 year market log return and 1 year T-bill rate as in Duffie et al. (2009) and other papers .

Remove terms

Df AIC
First model 12 7612.93
New denominator 12 7559.93
Add distance-to-default 13 7469.90
Add macro variables 15 7447.24
Simplify model 11 7441.04

Estimates – Standardized coefficients

Estimate Z-stat
Intercept -8.448 -17.985
Distance-to-default -1.929 -9.034
Log excess return -0.713 -18.711
Total liabilities / size 0.437 8.365
T-bill rate -0.281 -5.113
Net income / size -0.110 -5.650
Relative log market size -0.107 -1.702
Current ratio -0.101 -1.999
Idiosyncratic volatility 0.092 2.449
Log market return 0.078 2.287
Retained Earnings / size 0.042 1.543

Terms that are included in Duffie et al. (2009 table III) have similar sign.

Non-linear effects – Net income to size

A cubic regression spline is added to the model. Term on the linear predictor scale is on the y-axis. Dashed lines are 95% confidence intervals of the regression spline. A histogram of net income / size is shown in the background.

Non-linear effects – Idiosyncratic volatility

Significance of terms

AIC Df LR-stat P-value
Full model 7377.98
% Relative log market size 7378.14 1 2.16 0.141196
% Retained Earnings / size 7378.98 1 3.00 0.083270
% Log market return 7380.39 1 4.41 0.035750
% Current ratio 7382.43 1 6.45 0.011094
% Net income / size (spline term) 7393.20 3 21.23 0.000094
% Idiosyncratic volatility 7398.73 1 22.75 0.000002
% T-bill rate 7406.89 1 30.91 < 0.000001
% Idiosyncratic volatility (spline term) 7413.60 3 41.62 < 0.000001
% Distance-to-default 7418.65 1 42.67 < 0.000001
% Net income / size 7423.10 1 47.12 < 0.000001
% Total liabilities / size 7457.40 1 81.42 < 0.000001
% Log excess return 7732.36 1 356.38 < 0.000001

Rows are models without the given term. The spline terms are orthogonal to the linear term. P-values are based on likelihood ratio test.

In-sample results

Gray area is pointwise 95% confidence intervals. The line is the predicted distress rate. Dots are realised distress rates. Black dots are not covered by confidence intervals.

Out-of-sample results

Model is estimated up to the year prior to the forecasting year.

Out-of-sample results

Out-of-sample AUC. Black markers are from the model without non-linear effects and the blue markers are from the model with the covariates in Duffie et al. (2009) except log excess return is used instead of the return and the 1 year T-bill rate is used instead of the 3 month. The last markers are from the full model with splines.

Models with frailty

Starting point is a time-varying intercept as in Duffie et al. (2009).

\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, u_t)) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + u_t \\ u_t &\sim N(\theta u_{t - 1}, \sigma^2) \end{align*} \]

where \(g\) is the inverse complementary-log-log function.

It may be that do use \(\Delta t\) close to zero in their discrete approximation but it appears that they use \(\Delta t = 1\) month when you look at Appendix B and C. In that case, one could argue that a log-likelihood of Poisson process with a log-offset (the continuous version) should not be used but rather the inverse complementary-log-log function (the discrete version). It presumably does not matter.

Frailty model

Use an auxiliary particle filter and smoother to perform the estimation with a Monte Carlo EM or ECM algorithm. See Gordon, Salmond, and Smith (1993) for early work on particle filters, Pitt and Shephard (1999) for details on auxiliary particle filtering, and Fearnhead, Wyncoll, and Tawn (2010) for the smoother I use. See Dempster, Laird, and Rubin (1977) and Meng and Rubin (1993) for respectively details on the EM and ECM algorithm.

See the R package at CRAN.R-project.org/package=dynamichazard for implementation details.

Random intercept – Estimates

Df AIC
Model without frailty 17 7378
Random intercept 19 7345
Random intercept and log relative size 22 7229
Random intercept, log relative size, and industry dummies 25 7238

Estimates are \((\hat\theta, \hat\sigma) = (0.8912, 0.1484)\).

Random intercept – Smoothed Estimates

Dashed lines are 68.27% confidence intervals.

Random relative size – Motivation

Investigate reduced capital requirements of SMBs in Basel II/III as in Jensen, Lando, and Medhat (2017).

Under the IRB approach for corporate credits, banks will be permitted to separately distinguish exposures to SME borrowers (defined as corporate exposures where the reported sales for the consolidated group of which the firm is a part is less than €50 million) from those to large firms. A firm-size adjustment … is made to the corporate risk weight formula for exposures to SME borrowers.

Article 273 in Basel Committee on Banking Supervision (2006).

Random relative size – Model

\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, \vec u_t, o_{it})) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + u_{I,t} +u_{R,t} o_{it} \\ \begin{pmatrix} u_{I,t} \\ u_{R,t} \end{pmatrix} &\sim N\left( \begin{pmatrix} \theta_Iu_{I,t-1} \\ \theta_Ru_{R,t-1} \end{pmatrix}, Q\right) \end{align*} \]

Random relative size – Estimates

Df AIC
Model without frailty 17 7378
Random intercept 19 7345
Random intercept and log relative size 22 7229
Random intercept, log relative size, and industry dummies 25 7238

Estimates are

\[ \begin{align*} \begin{pmatrix} \hat\theta_I \\ \hat\theta_R \end{pmatrix} &= \begin{pmatrix} 0.9480 \\ 0.9793 \end{pmatrix} & \hat Q&= \hat V\hat C\hat V\\ \hat V&=\begin{pmatrix} 0.2446 & \cdot \\ \cdot & 0.0500 \end{pmatrix} & \hat C&=\begin{pmatrix} 1 & 0.9003 \\ 0.9003 & 1 \end{pmatrix} \end{align*} \]

Random relative size – Smoothed estimates

The fixed slope estimate 0.03975. The overall mean is subtracted from the relative log market size (and also from other firm specific covariates, \(\vec x_{it}\)).

Different from marginal effect

Points are means for distressed firms and crosses are the means of the whole sample. The two lines are smoothing splines.

Slope changes on T-bill rate

Model without frailty Model with frailty
Intercept -9.8216 -10.1538
Retained Earnings / size 0.0923 0.0850
Net income / size
Total liabilities / size 1.1879 1.1457
Current ratio -0.0979 -0.1366
Idiosyncratic volatility
Log excess return -1.3661 -1.3241
Relative log market size -0.0467 0.0397
Distance-to-default -0.3909 -0.3711
Log market return 0.4106 0.5064
T-bill rate -8.3157 -4.2765
Spline Net income / size
Spline Idiosyncratic volatility

Comparisson with other papers

Lando et al. (2013) find a time-varying log pledgeable assets effects in a non-parametric Aalen model.

Azizpour, Giesecke, and Schwenkler (2018) uses exponentially weighted log past defaulted debt in aggregate distress prediction.

Random industry – Motivation

Advocated by Chava and Jarrow (2004) and Chava, Stefanescu, and Turnbull (2011).

No evidence of industry effects in the model without frailty.

Use dummies from Chava and Jarrow (2004):

  • Manufacturing and mineral industries: SIC code in the ranges 1000–1499, 2000–3999.
  • Miscellaneous industries: SIC code in the ranges 1–999, 1500–1799, 5000–5999, 7000–8999.
  • Transportation, communications, and utilities: SIC code in the range 4000–4999
    •

Random industry – Model

\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, \vec u_t, \vec o_{it})) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + \vec u_{t}^\top\vec o_{it} \\ \begin{pmatrix} u_{I_1,t} \\u_{I_2,t} \\u_{I_3,t} \\ u_{R,t} \end{pmatrix} &\sim N\left( \begin{pmatrix} \theta_Iu_{I_1,t-1} \\ \theta_Iu_{I_2,t-1} \\ \theta_Iu_{I_3,t-1} \\ \theta_Ru_{R,t-1} \end{pmatrix}, Q\right) \end{align*}, \qquad Q=VCV \]

\[ V=\begin{pmatrix} \sigma_{I_1} & \cdot & \cdot & \cdot \\ \cdot & \sigma_{I_2} & \cdot & \cdot \\ \cdot & \cdot & \sigma_{I_3} & \cdot \\ \cdot & \cdot & \cdot & \sigma_R \end{pmatrix}, \qquad C = \begin{pmatrix} 1 & \rho_I & \rho_I & \rho_{IR} \\ \rho_I & 1 & \rho_I & \rho_{IR} \\ \rho_I & \rho_I & 1 & \rho_{IR} \\ \rho_{IR} & \rho_{IR} & \rho_{IR} & 1 \end{pmatrix} \]

Random industry – Estimates

Df AIC
Model without frailty 17 7378
Random intercept 19 7345
Random intercept and log relative size 22 7229
Random intercept, log relative size, and industry dummies 25 7238

\[ \begin{align*} (\hat\theta_I, \hat\theta_R) &= (0.9062, 0.9616) \\ (\hat\sigma_{I_1}, \hat\sigma_{I_2}, \hat\sigma_{I_3},\hat\sigma_R) &= (0.2998, 0.3484, 0.3049, 0.0607) \\ (\hat\rho_I, \hat\rho_{IR}) &= (0.9029, 0.9065) \end{align*} \]

The log-likelihood in the last model is actually smaller as is evident by the AIC. Though, it does seem like the EM is close to having converged in the latter model.

Random industry – Smoothed estimates

Comparisson with other papers

Compare with Kwon and Lee (2018).

  • No correlation of industry random effects and no random log relative size.
  • Different sample and questionable estimation method.

Compare with Chen and Wu (2014).

  • Only bankruptcies.
  • Japanese firms.
  • No random log relative size.

Conclusion

Document non-linear effects.

Rarely focused on. One example is Berg (2007).

Partial association between size of the firm and distress is non-constant through time.

Weak evidence for partial industry effects.

Next steps

Document importance. Could make model for \((\vec x_{it}, \vec z_t, \vec o_{it})\) as in Duffie, Saita, and Wang (2007), Duffie et al. (2009), Chen and Wu (2014), and Qi, Zhang, and Zhao (2014). This may lead to poor performance as documented in Duan, Sun, and Wang (2012).

Alternatively, forgo the the covariate prediction problem and make a 1 year distress model.

Next steps

Explore other time-varying coefficients.

Have no standard errors for effects. Compute observed information matrix with numerical differentiation or the methods in e.g., Cappé, Moulines, and Ryden (2005) or Poyiadjis, Doucet, and Singh (2011).

Potential issue with Moody’s data.

Thank you!

Slides are on github.com/boennecd/Talks.

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